Understanding Permutation and Combination -1

If I go back to my previous post of finding traveling salesman problem, and I need to know how many options are available to visit all the cities, understanding of permutations and combinations can help me.

In how many ways we can choose and arrange options. For example, you have to create N digit number as a code, say 4 digit number.

You can choose the number in 10^4 ways. This is how-

First digit- 10 options (0 to 9 – say 0 is allowed)
Second digit- 10 options
and so on.

So at the end we have
10*10*10*10 options or 10^4

or to generalize

n*n*n*n.. r times i.e. n^r

The above case is for permutation where repetition is allowed. There can be a case where repetition is not allowed, e.g. instead of choosing numbers randomly, we are picking random balls from a sack. So if you have picked ball number 7 once, it will not be available for reuse.

First digit- 10 options
Second digit- 9 options (first digit option not available for reuse).
Third- 8
Fourth- 7

Total Options 10*9*8*7


n*(n-1)*(n-2).. (n-r)

nPr = n!/(n-r)!